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Mobile Automaton with Memory

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A mobile automaton consists of a row of cells with two possible colors for each cell (black or white), with a single "active cell". The rule specifies the updated color for the active cell, based on its current color and that of its left and right neighbors. In addition to updating the color, the rule specifies whether the active cell should move right or left at each step. (To show where the active cell is at each time step, a black active cell is shown as a red cell, and a white active cell is shown as a blue cell.)

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This Demonstration lets you explore a mobile automaton with memory. Here, memory means the color of the active cell is taken not from the current step, but from a given number of time steps in the past. Memory zero is the regular mobile automaton case. With a memory of one, the color of the active cell is taken from the previous time step, and so on. The rule is summarized by the rule table below the time evolution. The memory is shown in an icon on the right-hand side of the rule table.

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Contributed by: Paul-Jean Letourneau and Paule Ecimovic (August 2011)
Open content licensed under CC BY-NC-SA

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The mobile automaton rule is controlled by separately setting the rules for the color update and the active cell movement (both independently range from 0 to 255), giving a total of possible rules. The memory can be chosen independently, from zero to 30, for any given rule.

A mobile automaton with memory requires an initial condition with rows.

Mobile automata with memory were first studied as abstract computational systems at the 2011 New Kind of Science Summer School. To date, nothing beyond purely repetitive (class 1) and nesting (class 2) behavior has been observed in two-color mobile automata, with or without memory (see page 71 of A New Kind of Science). That is, there are as yet no known cases where adding memory increases the complexity of a two-color mobile automaton rule. Adding memory does tend to cause interesting transients to occur, however. For example, an interesting transient occurs for rule {43,22} with memory 6, starting from a blank initial condition.

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